Quantum computation with quantum dots and terahertz cavity quantum electrodynamics

ABSTRACT

A quantum computer is proposed in which information is stored in the two lowest electronic states of doped quantum dots. Multiple quantum dots are located in a microcavity, and a pair of gates controls the energy levels in each quantum dot. A controlled NOT (CNOT) operations involving any pair of quantum dots can be effected by a sequence of gate voltage pulses which tune the quantum dot energy levels into resonance with frequencies of the cavity or a laser. The duration of a CNOT operation is estimated to be much shorter than the time for an electron to decohere by emitting an acoustic phonon.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is related to Provisional Patent Application Ser. No.60/112,439, filed Dec. 16, 1998, entitled “QUANTUM COMPUTATION WITHQUANTUM DOTS AND TERAHERTZ CAVITY QUANTUM ELECTRODYNAMICS,” by Mark S.Sherwin et al., and also related to Provisional Patent Application Ser.No. 60/123,220, filed Mar. 8, 1999, entitled “QUANTUM COMPUTATION WITHQUANTUM DOTS AND TERAHERTZ CAVITY QUANTUM ELECTRODYNAMICS,” by Mark S.Sherwin et al, which applications are incorporated by reference herein.This application also claims priority under 35 U.S.C. § 119(e) to bothProvisional Patent Application Ser. No. 60/112,439, filed Dec. 16, 1998,entitled “QUANTUM COMPUTATION WITH QUANTUM DOTS AND TERAHERTZ CAVITYQUANTUM ELECTRODYNAMICS,” by Mark S. Sherwin et al. and ProvisionalPatent Application Ser. No. 60/123,220, filed Mar. 8, 1999, entitled“QUANTUM COMPUTATION WITH QUANTUM DOTS AND TERAHERTZ CAVITY QUANTUMELECTRODYNAMICS,” by Mark S. Sherwin et al.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH AND DEVELOPMENT

This invention was made with Government support under Grant No. ARODAAG55-98-1-0366, awarded by the Army. The Government has certain rightsin this invention.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates in general to quantum computation, and inparticular to quantum computation with quantum dots and terahertz cavityquantum electrodynamics.

2. Description of Related Art

A quantum computer processes quantum information which is stored in“quantum bits,” also called qubits. If a small set of fundamentaloperations, or universal quantum logic gates, can be performed on thequbits, then a quantum computer can be programmed to solve an arbitraryproblem. Quantum computation has been shown to efficiently factorizelarge integers, and the quantum information can be stored indefinitely,which provides the interest in quantum computation and machines that canperform quantum computation.

Consider, for example, the publication by Barenco, et al., entitled“Conditional Quantum Dynamics In Logic Gates,” Physical Review Letters,15 May 1995, USA, vol. 74, no. 20, pages 4083–4086. This publicationnotes that quantum logic gates provide fundamental examples ofconditional quantum dynamics, and could form the building blocks ofgeneral quantum information processing systems, which have recently beenshown to have many interesting non-classical properties. Thispublication describes a simple quantum logic gate, the quantumcontrolled-NOT (CNOT), and analyzes some of its applications. Thepublication also discusses two possible physical realizations of thegates, one based on Ramsey atomic interferometry, and the other on theselective driving of optical resonances of two subsystems undergoing adipole—dipole interaction.

However, the implementation of a large-scale quantum computer hasremained a technological challenge. The qubits must be well isolatedfrom the influence of the environment, but must remain manipulatable inindividual units to initialize the computer, perform quantum logicoperations, and measure the result of the computation.

Implementations of such a quantum computer have been proposed usingatomic beams, trapped atoms and/or ions, bulk nuclear magneticresonance, nanostructured semiconductors, and Josephson junctions.However, each scheme proposed has limitations that make large-scaleimplementation difficult and very limiting in performance.

For example, proposals using trapped atoms or ions, qubits couple withcollective excitations or cavity photons. This coupling enables two-bitgates involving an arbitrary pair of qubits which makes programmingstraightforward. However, these schemes require serial gating schemes,whereas error correction schemes require parallelism, thereby limitingthe usefulness of data gathered using an atomic or ion trapping machine.

In the semiconductor and superconductor schemes, only nearest-neighborqubits can be coupled, and significant overhead is required to coupledistant qubits. However, these machines can perform some gate operationsin parallel, which allows for some error correction.

It can be seen, then, that there is a need in the art for a quantumcomputer. It can also be seen, then, that there is a need in the art fora quantum computer that can perform parallel gate operations. It canalso be seen, then, that there is a need in the art for a quantumcomputer that can perform parallel gate operations without significantqubit overhead.

SUMMARY OF THE INVENTION

To overcome the limitations in the prior art described above, and toovercome other limitations that will become apparent upon reading andunderstanding the present specification, the present invention disclosesan apparatus and method for quantum computing. The apparatus comprises acontrol bit structure, a target bit structure, and gate electrodes,coupled to the control bit structure and the target bit structure, forapplying a voltage across the control bit structure and the target bitstructure, wherein the control bit structure and the target bitstructure obtain quantum levels of excitation from the applied voltagesbased on an initial excitation level of the control bit structure and aninitial excitation level of the target bit structure.

The method of the present invention comprises applying a first voltageacross a control bit structure, applying a second voltage across atarget bit structure, and obtaining quantum levels of excitation withinthe control bit structure and the target bit structure based on theapplied first and second voltages, an initial excitation level of thecontrol bit structure and an initial excitation level of the target bitstructure.

An object of the present invention is to provide a quantum computer.Another object of the present invention is to provide a quantum computerthat can perform parallel gate operations. A further object of thepresent invention is to provide a quantum computer that can performparallel gate operations without significant qubit overhead.

These and various other advantages and features of novelty whichcharacterize the invention are pointed out with particularity in theclaims annexed hereto and form a part hereof. However, for a betterunderstanding of the invention, its advantages, and the objects obtainedby its use, reference should be made to the drawings which form afurther part hereof, and to the accompanying detailed description, inwhich there are illustrated and described specific examples of a methodand apparatus in accordance with the invention.

BRIEF DESCRIPTION OF THE DRAWINGS

Referring now to the drawings in which like reference numbers representcorresponding parts throughout:

FIG. 1 illustrates the computer of the present invention;

FIG. 2 illustrates the potential and four lowest electronic energylevels for a particular realization of a quantum dot within the computerof the present invention;

FIG. 3A illustrates the energy levels of the transitions in a quantumdot computer of the present invention;

FIG. 3B illustrates a sequence of voltage pulses that effect a two-qubitgate that is equivalent to a CNOT operation in a quantum dot computer ofthe present invention;

FIG. 4 illustrates a schematic diagram of a quantum computer of thepresent invention using quantum dot spins coupled to a cavity which isnear resonance with an intraband transition in the quantum dots;

FIG. 5 illustrates the energy level structure, couplings, and detuningswithin a quantum bit of the computer of FIG. 4;

FIG. 6 illustrates an illustration of a readout of a quantum bit in thespin-state computer of the present invention; and

FIG. 7 is a flow chart illustrating the steps used to practice thepresent invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

In the following description of the preferred embodiment, reference ismade to the accompanying drawings which form a part hereof, and in whichis shown by way of illustration a specific embodiment in which theinvention may be practiced. It is to be understood that otherembodiments may be utilized and structural changes may be made withoutdeparting from the scope of the present invention.

Overview

A quantum computer of the present invention stores information in thetwo lowest quantum electronic states of doped quantum dots. Multiplequantum dots are located in a microcavity, and a pair of gates controlsthe energy levels in each quantum dot. A controlled NOT (CNOT)operations involving any pair of quantum dots can be effected by asequence of gate voltage pulses which tune the quantum dot energy levelsinto resonance with frequencies of the cavity or a laser. The durationof a CNOT operation is estimated to be much shorter than the time for anelectron to decohere by emitting an acoustic phonon.

Quantum Bits and Fundamental Quantum Logic Operations

FIG. 1 illustrates the computer of the present invention. The computercomprises two or more structures 100 embedded in a microcavity 126. Eachstructure 100 is comprised of a control nanostructure 102 and a targetnanostructure 104. The control nanostructure 102 controls the controlbit 102 of the computer, and the target nanostructure 104 controls thetarget bit 104 of the computer. Although described separately, controlnanostructure 102 and target nanostructure 104 are substantially similarand interchangable within the computer.

Each nanostructure 102 and 104 comprises outer semiconductor layers 106and 108 and disks 110–114. Although three disks 110–114 are shown, alarger or smaller number of disks 110–114 can be used without departingfrom the scope of the present invention. The disks 110–114 are typicallysmaller in bandgap than the outer semiconductor layers 106–108, e.g., ifthe disks 110–114 are GaAs, then the outer semiconductor layers are of alarger band gap material than GaAs, e.g., AlGaAs. The central disk 112is typically larger or taller than the outer disks 110 and 114. Thebarrier 116 between disks 110 and 112 and the barrier 118 between disks112 and 114 are sufficiently thin to allow an electron to rapidly tunnelthrough the barriers 116 and 118. A structure consisting of a set ofthree disks 110–114 and the two intervening barriers 116–118 ishereafter called a quantum dot (QD), referred to herein as control bit102 and target bit 104. Each QD 102 and 104 that participates in thequantum computation must have one and only one electron within the QD102 or 104.

Below and above each QD 102 and 104 is an electrical gate 120 and 122,shown in QDs 102 and 104. These gates are used to apply electricalvoltages substantially simultaneously to QD 102 and 104 across thelength 124 of a QD 102 or 104. The QDs 102 and 104 are located in athree-dimensional cavity 126. The cavity 126 can contain many QDs 102and 104.

FIG. 2 illustrates the potential and four lowest electronic energylevels for a particular realization of a QD within the computer of thepresent invention. The lowest two energy levels 200 and 202, alsoreferred to as energy levels |1> and |1>, form the qubits which storequantum information. The third energy level 204, also referred to asenergy level |2>, serves as an auxiliary state to perform conditionalrotations of the state vector of the qubit. Voltages applied to thegates 120 and 122 are used to control the spacing between and absolutevalues of the energy levels of the QDs 102 and 104 via the Stark effect.As shown in FIG. 2, the energy levels 200, 202, 204 and 206 are allbelow the barrier heights of barriers 116 and 118, and are allowedenergy states in each of the disks 110–114. A large number ofindividually-gated QDs 102 and 104 are contained in a 3-D microcavity126 whose fundamental resonance has a wavelength λ_(C) much longer thanthe length of a QD 124. A continuous-wave laser with a fixed wavelengthdifferent than λ_(C) is introduced through one side of the cavity 126.

FIG. 3A illustrates the energy levels of the transitions in a quantumdot computer of the present invention, and FIG. 3B illustrates asequence of voltage pulses that effect a two-qubit gate that isequivalent to a CNOT operation in a quantum dot computer of the presentinvention.

With reference to FIG. 3A, energies E₁₀ 300 and E₂₀ 302 show the state 0to state 1 and state 0 to state 2 energy levels, respectively. Theenergies of a cavity 126 mode photon hω_(C), 304, a laser photon hω_(L)306, and the sum of hω_(C)+hω_(L) 308, are also shown. The state of anelectron in a QD 102 or 104 can be coherently manipulated by tuning E₁₀300 and E₂₀ 302 into and out of resonance with

ω_(C), 304, hω_(L) 306, and the sum of hω_(C)+

ω_(L) 308.

A general Hamiltonian describing a QD 102 or 104 interacting with cavity126 photons and the laser field is given byĤ=

ω _(C) â _(C) +E ₁₀(e)σ₁₁ +E ₂₀(e)σ₂₂+

g ₀₁(e){â _(C)+σ₀₁+σ₁₀ +â _(C)}+

Ω_(1,01)(e){σ₀₁ exp(iω ₁ t)+σ₁₀ exp(iω ₁ t)}+

₁₂(e){â _(C)+σ₁₂+σ₁₂ â _(C)}+

Ω_(1,12)(e){â _(C)σ₁₂ exp(iω ₁ t)+σ₂₁ â _(C) exp(ωt)}

-   -   where â_(C) denotes the cavity 126 mode annihilation operator,        and    -   σ_(IJ)=|i×j| is the projection operator from QD state |j> to        state |i>.

The vacuum Rabi frequencies are g_(IJ)≈qz_(IJ)e_(VAC),

-   -   where $e_{VAC} = \sqrt{\frac{{\hslash\omega}_{C}}{2ɛ_{0}ɛ\; V}}$    -    is the amplitude of the vacuum electric field in the cavity        126,    -   ε=the dielectric constant of the cavity 126,    -   V=the volume of the cavity 126,    -   q=the electronic charge, and    -   z_(IJ) is the dipole matrix of the |i> to |j> transition.

One step in the CNOT operation is a Rabi oscillation between states |0>and |2> involving both cavity 126 and laser photons at e=e_(L+C).

Operation of the Quantum Computer

During the operation of the quantum computer of the present invention, aqubit that stores quantum information is in state |0> or state |1>, andthe electric field across the qubit is held at a value where the energylevels of the qubit are not resonant with

ω_(C),

ω_(L), or

ω_(C)+

ω_(L). The value of this electric field is typically zero, but can beother values. The typical value of the electric field is called thefiducial value of the electric field. For e≈e_(C), and either the cavity126 contains one photon or the qubit state vector is in state |1>, thenthe qubit will execute vacuum Rabi oscillations with frequency g₀₁₄, inwhich the probability of finding one photon in the excited stateoscillates ninety degrees out of phase with the probability of findingone photon in the cavity 126. For e≈e_(L), the state vector of the qubitrotates between states |0> and |1> with laser Rabi frequency Ω_(1,01).For e≈e_(L+C), and the cavity 126 contains one photon and the qubitstate vector begins in state |0>, then the qubit rotates between states|0> and the auxiliary state |2> with frequency Ω(e_(L+C)). If either thequbit is in state |1> or the cavity 126 does not contain a photon, thenthe qubit state vector is not rotated for e≈e_(L+C).

The Controlled NOT Operation

A Controlled NOT (CNOT) operation is effected by a series of voltagepulses applied across the gates of a pair of qubits. The pulses beginand end with the qubit at the fiducial electric field (e=0), and rise toa target value of e_(C), e_(L), or e_(L+C).

With reference to FIG. 3B, the cavity 126 always begins without anyphotons. First, a “π” pulse 310 with height e_(C) 312 and durationπ/2g₀₁, 314 is applied to the control bit 102, e.g., across contacts 120and 122 of QD 102. If the control bit 102 is in state |0>, the controlbit 102 is unaffected. If the control bit 102 is in state |1>, thecontrol bit 102 rotates into state |0> and acquires a phase i, and thecavity 126 acquires a single photon.

A 2π pulse 316 with height e_(L+C) 318 and duration π/Ω(e_(L+C)) 320 isthen applied to the target bit 104. If the target bit 104 is in state|1>, the target bit 104 is unaffected. If the target bit 104 is in state|0>, and the cavity 126 contains one photon, the target bit 104 acquiresa phase of −1.

A pulse 322 with height e_(C), substantially identical to the π pulse310, is again applied to the control bit 102. If there is a photon inthe cavity 126 it is absorbed by the control bit 102, returning thecontrol bit 102 to state |1> while the control bit 102 acquires anotherphase i. The end result is a gate in which the state vector of thetwo-qubit system, e.g., the two qubits being the control bit 102 and thetarget bit 104, acquires a phase −1 if and only if both control andtarget bits 102 and 104 are initially 1.

The sequence of state-vector rotations which is effected by the seriesof electric field pulses is identical to the sequence effected by aseries of laser pulses applied to cold trapped ions. In order to effecta CNOT operation, i.e., inversion of the target bit 104 if and only ifthe control bit 102 is 1, it is necessary to apply to the target bit 104“π/2” and “3π/2” pulses with height e_(L) and durations π/(4Ωe_(L+C))and 3π/(4Ω_(L,01)), respectively, before and after the sequence shown inFIG. 3B.

In essence, the electric field pulses applying a first voltage across acontrol bit structure and a second voltage across a target bitstructure. The quantum levels of excitation within the control bitstructure and the target bit structure are obtained based on the appliedfirst and second voltages, an initial excitation level of the controlbit structure, and an initial excitation level of the target bitstructure. As described above, the target bit structures and the controlbit structure are interchangeable within the present invention, e.g.,for a first computation, a first structure can be the control bitstructure and a second structure can be the target bit structure. For asecond computation, the first structure can be the target bit structureand the second structure can be the control bit structure.

To ensure the fidelity of CNOT operations, the rise and fall times ofthe pulses 310, 316, and 322 must be short compared to the period of theRabi oscillation at the target bit 104 electron. At the same time, inorder to minimize the probability of a transition between the |0> and|1> states induced by the ramping electric field, the changes to theHamiltonian must be adiabatic, e.g., δt>>

/E₁₀. Further, the timing between the successive pulses 310, 316, and322 in the CNOT operation must be adjusted to compensate for thequantum-mechanical phases accumulated by inactive qubits in theirexcited states. It also may be required to adjust the heights anddurations of the pulses 310, 316, and 322 to account for alternatingcurrent Stark shifts in the energy levels of the QDs 102–104 which areinduced by the laser field.

Other Actions Performed on the Quantum Computer

Other actions that are performed or required by a quantum computerinclude initialization of the computer, inputting data, reading out thedata stored in the computer, correcting errors in the computer, anddecoherence of the electronic state of the QD 102 and 104.

Initialization of the quantum computer requires that each qubit 102–104be in a well-defined state prior to any quantum computation. The presentinvention performs initialization by applying the proper fiducialvoltage to the gates 120–122 of the qubits 102–104, and a propertemperature to each qubit 102–104 for the requisite amount of time,until each qubit 102–104 relaxes to a ground state. Typically, a voltageof 0 volts, a temperature of T<120 Kelvin, and a wait of less than onesecond ensures that all qubits 102–104 are in state |0>.

To input initial data, arbitrary rotations of the state vectors of thequbits 102–104 are required to load data into the qubit 102–104registers. Arbitrary one-bit rotations of the qubits 102–104 areeffected in the present invention by using Rabi oscillations induced bythe laser field, by applying pulses with height e_(L) and durationbetween 0 and 2π/(Ω_(L,01)). To read the data back from the qubit102–104 registers after a quantum computation is completed, the state ofeach qubit 102–104 is measured. A narrow-band detector with high quantumefficiency is used to detect single terahertz (THz) photons at thefrequency of the cavity 126 mode ω_(C). The qubits 102–104 can then beread out sequentially by tuning each qubit 102–104 to ω_(C). If thequbit 102–104 is in state |1>, it will emit a photon which will bedetected by the narrow-band detector. If the qubit 102–104 is in state|0>, no photon will be emitted. The emissions and non-emissions fromeach qubit 102–104 can then be read by the detector and reported.

For error correction, the qubits 102–104 must be executed in parallel.To perform parallel execution, the cavity 126 is enlarged to createseveral cavity 126 modes in the frequency range over which the QD102–104 energy level spacings are tunable. A separate but equivalentapproach is to couple nearest-neighbor QDs 102–104, and perform anenlarged cavity 126 schema as in the present invention.

Decoherence

Decoherence of the electronic state of the QD 102–104 as well asdecoherence of the cavity 126 photons are problematic areas for quantumcomputers 100. Deductions on the energy relaxation times result in timesof q/I=10⁻⁷ seconds for transitions with energies near 50 microelectronvolts (μeV). However, the geometry of the present invention is quitedifferent, and, as such, results in different relaxation times. Thelifetime of a cavity 126 photon must be sufficiently long to enable manyNOT operations with high fidelity. As such, the cavities must below-loss, few-mode THz cavities.

Decoherence and CNOT Performance Times

Consider now a specific GaAs/AlGa is QD and lossless dielectric cavity126 designed to minimize the time required for a CNOT operation, whileat the same time avoiding the emission of longitudinal optical (LO)phonons (

ω_(LO)≈36 meV in GaAs) and also minimizing the rate of acoustic phononemission. Cavity 126 and laser photon energies are chosen to be 11.5 and15 meV. These energies are sufficiently large to enable an adequatevacuum electric field e_(VAC) while their sum is still comfortablysmaller than

ω_(LO). Assuming perfect cylindrical symmetry of the disks 110–114, thestates 200–206 are labeled with quantum numbers |1,m,n>, associated withthe radial, azimuthal and axial degrees of freedom, respectively.

The potential along the cylindrical axis of the QD (z-direction) 102 andthe numerically-computed four lowest energy levels 200–206 are depictedin FIG. 2. FIG. 3A shows the transitions E₁₀ and E₂₀ vs. electric fielde. Assuming infinite walls in the radial direction, the radialwavefunctions are given by Bessel functions. The difference between theenergy of the ground state 200 and first radial excited state 202 isΔE_(R)=30 meV for radius of the disks 110–114 of a=13 nm, assumingm*=m_(C)/15. This is higher than the highest energy reached by anelectron during a CNOT operation (26.5 mEv=

ω₁+

ω_(C)), eliminating dccoherence arising from coupling between axial andradial excited states of the QD 102–104.

The time required to execute a CNOT operation for the QD structure ofFIG. 1 is estimated at 150 microseconds. Unconditional 1-bit rotationswhich occur at e=e_(L) take only a few picoseconds for a laser electricfield of 30 kV/m. Although the laser might need attenuation for suchrotations to have a transition time for the electrical pulse shorterthan the period of the Rabi oscillation at the target electric field,the decoherence times and transition times allow the computer to performseveral thousand CNOT operations before decoherence occurs.

Additional Embodiment of the Quantum Computer

The present invention can also be embodied as a quantum dot doped with asingle electron. The spin-states of this conduction-band electron canserve as a qubit with long coherence times.

FIG. 4 illustrates a schematic diagram of a quantum computer of thepresent invention using quantum dot spins coupled to a cavity which isnear resonance with an intraband transition in the quantum dots.

Disk 400 is a whispering gallery mode resonator for terahertz radiation,typically fabricated from an undoped semiconductor material, such assilicon or gallium arsenide. Quantum dots 402–426 are embedded in disk400. Each quantum dot 402–426 contains a single electron. Each quantumdot 400–426 has an intraband transition which is close to the resonantfrequency of the same single mode of the whispering gallery moderesonator disk 400. For an alternative implementation, a magnetic field428 can be used. Laser beams 430–436 are focused on quantum dots402–426, such that a set of laser beams 430–432 are focused on eachquantum dot 402–426. For ease of illustration, only laser beams 430–436are shown, but each quantum dot 402–426 has a set of laser beams, e.g.,L_(n) ^(a), L_(n) ^(b), L_(n) ^(c), . . . for quantum dot “n” 402–426.Each laser beam 430–436 has a frequency and intensity which can beadjusted independently of the other laser beams 430–436. The laser beams430–436 are used to effect one- and two-bit quantum logic gates from thequantum dots 402–426. Alternatively, the cavity of disk 400 can beembodied as a defect in a photonic bandgap structure instead of awhispering gallery mode resonator, or in a superconductor. The cavitystructure shown in FIG. 4 illustrates a spin-terahertz cavity quantumelectron dot structure 438. Although shown in a substantially horizontalorientation, quantum dots 402–426 can be coupled in any direction, e.g.,horizontal, vertical, or any combination of horizontal and verticalorientations. Each quantum dot 402–426 can also have internal structuressimilar to that described in FIG. 1, thereby making each quantum dot402–426 a quantum bit in a quantum computer.

This structure 438 allows for a spin-splitting of the ground-statequantum dot 402–426 conduction band electron, using either a non-zeromagnetic field 428 across disk 400, or a pseudo-magnetic field generatedusing an off-resonant circularly polarized laser beam 430–436. Whenusing laser beams 430–436, large effective magnetic fields are used tointroduce large effective magnetic fields yielding a spin-splitting thatcan be as large as 5 meV.

Further, one bit rotations of a single quantum dot 402–426 electron spincan simply be achieved by spin-flip near-resonant Raman transitions viaintermediate valence-band states. The large spin-orbit coupling in thevalence band enables coherent flipping of the electron spin in shorttime scales using two laser beams, e.g., 430–432 for quantum dot 402,with orthogonal linear polarizations that can realize π/r pulses, wherer is any real number. If the spin splitting is generated by the ac-starkeffect of laser beams 430–436 rather than a magnetic field 428, then thespin-flipping can be accomplished by irradiating the entire sample withan oscillating magnetic field, and using the ac stark effect to tune thespin-splitting of selected quantum dots into resonance with theoscillating magnetic field for a duration long enough to effect π/rpulses.

Structure 438 allows for a cavity mode that has an energy thatcorresponds to an intra-band energy spacing. The advantage of structure438 and structure 100 over other embodiments using this terahertzcavity-quantum electron dot regime is that the wavelength of the cavitymode, which in turn determines the length scale of the quantum dot402–426 array, could exceed 100 microns, which allows for a large numberof quantum dots 402–426 to be coupled through the same cavity mode.

FIG. 5 illustrates the energy level structure, couplings, and detuningswithin a quantum bit of the computer of FIG. 4.

To effect non-trivial two-qubit interactions, structure 438 usesselective introduction of a transverse spin-spin coupling between twodistant quantum dots. However, structure 438 allows for a coherent drivethat couples the ground state 500 and excited state 502 with oppositespin in a single quantum dot 402–426. Further, the cavity mode couplesground state 500 and excited state 502 with the same spin. The coherentdrive at a particular quantum dot n 402 and the cavity mode are detunedby an energy 504, shown as δ_(c-c) ^(n). Together, the coherent driveand the cavity coupling provide a Raman coupling between spin up andspin down in the quantum dot 502, separated by an energy 506 δ_(spin)^(n). The detuning of the Raman transition in a particular dot from thespin-flip transition is Δ^(n)=δ_(c-c) ^(n)−δ_(spin) ^(n). Distantquantum dots with the same detuning Δ experience an effective 2 qubitinteraction which leads to controlled entanglement in general and CNOToperations in particular. CNOT operations between pairs of quantum dots402 and 404 with different shared detunings can thus proceed in parallelwith the present invention. For example, if quantum dots 402 and 404share detuning Δ_(a), and quantum dots 406 and 408 share a detuningΔ_(b) not equal to Δ_(a), the CNOT operations involving quantum dots 402and 404 can proceed in parallel with the CNOT operations involving dots406 and 408.

The coherent drive can be implemented in a variety of ways. For example,two of the laser beans 430 and 432, with frequencies differing byω_(coherent drive), shown as difference 508 in FIG. 5, can be applied toeach individual quantum dot 402–426. In this case, the coherent couplingis enhanced if the shape of the quantum dot 402–406 is asymmetric. Analternative embodiment of implementing coherent coupling is via acoherent terahertz field and a magnetic field 428 that is perpendicularto the effective magnetic field induced by a circularly polarized laserbeam 430.

The method described with respect to the present invention is useful inimplementing a two-qubit operation, like a CNOT operation, betweendistant spins embedded in a cavity which is resonant with an intrabandtransition.

One method is to set the real magnetic field 428 B=0. Two laser beams430 and 432 are used, e.g., L_(m) ^(a) and L_(m) ^(b) and are incidenton quantum dot 402, while a second pair of laser beams 434 and 436 areincident on quantum dot 404. One of these laser fields, e.g., L_(m)^(a), is circularly polarized, and determines the spin splitting of theground state of quantum dot 402, via the ac Stark effect. The secondlaser field focused on quantum dot 402, L_(m) ^(b), is detuned from thefrequency of L_(m) ^(a) by ω_(coherent drive), providing an effectivecoherent drive. When the two-photon detunings Δ are chosen, they aredetermined by the energy difference between spin splitting and theenergy difference of the cavity mode and the coherent drive and areidentical for the control and target qubits. Transverse spin-spincoupling can thus be established. Such coupling can implement a CNOTgate. One advantage of this particular implementation is that theenergies of the spin states in a quantum dot 402–426 are different onlywhile the lasers 430–436 are turned on. While the lasers 432–436 areoff, no quantum-mechanical phase difference between ground state 500 andexcited state 502 will accumulate.

Another method to implement the two-qubit operation is to set anexternal magnetic field 428 B=B_(x) where field 428 is substantiallyperpendicular to the effective magnetic field induced by the circularlypolarized laser beam 430. For example, in disk-shaped quantum dots402–426 with strong confinement in the z-direction, a circularlypolarized laser field 430 is applied that generates an effectivemagnetic field in the z-direction. A coherent terahertz field is appliedthat is polarized parallel to the cavity mode. In such a configuration,parallel linearly polarized coherent terahertz and cavity modes withenergy difference near the ground-state spin-splitting can be used toachieve the necessary coupling between the two spin states.

FIG. 6 illustrates an illustration of a readout of a quantum bit in thespin-state computer of the present invention. The states of the quantumbits 402 and 404 can be read by using resonant fluorescence of thequantum bits 402 and 404. A magnetic field 428 is applied to disk 400,which splits the spin states of the electron and the hole in eachquantum dot 402–426. A circularly polarized laser beam 430 is tuned to atransition between the valence band and one of the spin states 600–606in the quantum dot 402–426. If the spin state is empty, e.g., states 600and 602 in quantum dot 404, the laser 430 field alternatively populatesand stimulates emission from the empty state 602, which results in aresonance fluorescence signal. The resonance fluorescence signal lastsfor as long as the state 602 remain empty. If the state 602 is occupiedby an electron, e.g., as shown in quantum dot 402, the absorption of thelaser 420 field is Pauli blocked, and no light is emitted from thequantum dot 402. The emission/lack of emission provides a readout of thestate of quantum dots 402–426.

Process Chart

FIG. 7 is a flow chart illustrating the steps used to make the quantumdots of the present invention.

Block 700 illustrates performing the step of growing a first quantum dotlayer on an edge layer.

Block 702 illustrates performing the step of growing a first barrierlayer on the first quantum dot layer.

Block 704 illustrates performing the step of growing a second quantumdot layer on the first barrier layer.

Block 706 illustrates performing the step of growing a second barrierlayer on the second quantum dot layer.

Block 708 illustrates performing the step of growing a third quantum dotlayer on the second barrier layer.

Block 710 illustrates performing the step of growing a second edge layeron the third quantum dot layer wherein the edge layer, first quantum dotlayer, first barrier layer, second quantum dot layer, second barrierlayer, third quantum dot layer, and second edge layer comprise at leastone bit in the quantum computer.

To grow the quantum dots of the present invention as shown in FIG. 1,stacked self-assembled quantum dots can be used. Another method is tomake QDs made by growing GaAs/AlGaAs quantum wells with the conductionband profile tailored to give the desired potential in the z-direction,depositing small islands on top of the quantum well to serve as an etchmask, etching through the quantum well layers which are not protected bythe islands, and then regrowing AlGaAs. The growth methods used to growthe QDs 102 and 104 comprise those used in the art, such as MetalOrganic Chemical Vapor Deposition (MOCVD), Metal Organic Molecular BeamEpitaxy (MOMBE), wet or dry etching of the materials, or other growthmethods.

CONCLUSION

In summary, the present invention discloses an apparatus and method forquantum computing. The apparatus comprises a control bit structure, atarget bit structure, and gate electrodes, coupled to the control bitstructure and the target bit structure, for applying a voltage acrossthe control bit structure and the target bit structure, wherein thecontrol bit structure and the target bit structure obtain quantum levelsof excitation from the applied voltages based on an initial excitationlevel of the control bit structure and an initial excitation level ofthe target bit structure.

The method of the present invention comprises applying a first voltageacross a control bit structure, applying a second voltage across atarget bit structure, and obtaining quantum levels of excitation withinthe control bit structure and the target bit structure based on theapplied first and second voltages, an initial excitation level of thecontrol bit structure and an initial excitation level of the target bitstructure.

The foregoing description of the preferred embodiment of the inventionhas been presented for the purposes of illustration and description. Itis not intended to be exhaustive or to limit the invention to theprecise form disclosed. Many modifications and variations are possiblein light of the above teaching. It is intended that the scope of theinvention be limited not by this detailed description, but rather by theclaims appended hereto.

1. A method for effecting gate operations using one or more semiconductor quantum bits, wherein the semiconductor quantum bits are contained in a cavity, an electromagnetic field is applied to excite the semiconductor quantum bits to one or more energy levels, and the semiconductor quantum bits so excited contain information used to implement the gate operations, the method comprising: coherently coupling the semiconductor quantum bits using a mode in the cavity that has a resonant frequency substantially coincident with a transition between the energy levels of the semiconductor quantum bits.
 2. The method of claim 1, wherein the semiconductor quantum bits are arranged in an array.
 3. The method of claim 1, wherein each of the semiconductor quantum bits is a quantum dot doped with a single electron.
 4. The method of claim 3, wherein the information is represented by the states of an electron trapped in the quantum dot.
 5. The method of claim 4, wherein the information is contained in a spin-state of the electron.
 6. The method of claim 5, further comprising the step of reading the information by determining the spin-state of the trapped electron.
 7. The method of claim 6, wherein the spin-state is determined by detecting selective fluorescent emissions from the trapped electron.
 8. The method of claim 1, wherein the cavity is a whispering-gallery-mode resonator.
 9. The method of claim 1, wherein the cavity is a defect in a photonic band-gap structure.
 10. The method of claim 1, wherein the cavity is a superconductor structure.
 11. The method of claim 1, wherein the electromagnetic field includes a component generated by one or more laser beams.
 12. The method of claim 1, wherein the electromagnetic field includes a component generated by an externally applied magnetic field.
 13. The method of claim 1, wherein the gate operations result in a conditional NOT operation.
 14. The method of claim 1, wherein the semiconductor quantum bits are vertically coupled quantum dots.
 15. The method of claim 1, wherein the semiconductor quantum bits are horizontally coupled quantum dots.
 16. A quantum computing apparatus, comprising: a cavity containing one or more semiconductor quantum bits; and means for applying an electromagnetic field to the cavity to excite the semiconductor quantum bits to one or more energy levels, wherein the semiconductor quantum bits are coherently coupled using a mode in the cavity that has a resonant frequency substantially coincident with a transition between the energy levels of the semiconductor quantum bits.
 17. The apparatus of claim 16, wherein the semiconductor quantum bits are arranged in an array.
 18. The apparatus of claim 16, wherein each of the semiconductor quantum bits is a quantum dot doped with a single electron.
 19. The apparatus of claim 18, wherein the information is represented by the states of an electron trapped in the quantum dot.
 20. The apparatus of claim 19, wherein the information is contained in a spin-state of the electron.
 21. The apparatus of claim 20, further comprising means for reading the information by determining the spin-state of the trapped electron.
 22. The apparatus of claim 21, wherein the spin-state is determined by detecting selective fluorescent emissions from the trapped electron.
 23. The apparatus of claim 16, wherein the cavity is a whispering-gallery-mode resonator.
 24. The apparatus of claim 16, wherein the cavity is a defect in a photonic band-gap structure.
 25. The apparatus of claim 16, wherein the cavity is a superconductor structure.
 26. The apparatus of claim 16, wherein the electromagnetic field includes a component generated by one or more laser beams.
 27. The apparatus of claim 16, wherein the electromagnetic field includes a component generated by an externally applied magnetic field.
 28. The apparatus of claim 16, wherein the quantum computing apparatus performs gate operations that result in a conditional NOT operation.
 29. The apparatus of claim 16, wherein the semiconductor quantum bits are vertically coupled quantum dots.
 30. The apparatus of claim 16, wherein the semiconductor quantum bits are horizontally coupled quantum dots.
 31. A method of storing information in quantum states of electrons in semiconductor quantum bits comprising electron-doped quantum dots, wherein multiple quantum dots are located in a cavity excited by an electromagnetic field, the method comprising: effecting a controlled NOT (CNOT) operation involving any pair of quantum dots by tuning energy levels of the quantum dots into resonance with frequencies of the cavity.
 32. The method of claim 31, wherein the energy levels of the quantum dots are tuned by voltages applied to gates across the quantum dots.
 33. The method of claim 31, wherein the energy levels of the quantum dots are tuned by pulses of electromagnetic radiation focused onto the quantum dots.
 34. The method of claim 31, wherein the semiconductor quantum bits are arranged in an array.
 35. The method of claim 31, wherein each of the semiconductor quantum bits is a quantum dot doped with a single electron.
 36. The method of claim 35, wherein the information is represented by the states of an electron trapped in the quantum dot.
 37. The method of claim 36, wherein the state of the electron is determined by detecting selective fluorescent emissions from the trapped electron.
 38. The method of claim 36, wherein the information is contained in a spin-state of the electron.
 39. The method of claim 38, further comprising the step of reading the information by determining the spin-state of the trapped electron.
 40. The method of claim 39, wherein the spin-state is determined by detecting selective fluorescent emissions from the trapped electron.
 41. The method of claim 31, wherein the cavity is a whispering-gallery-mode resonator.
 42. The method of claim 31, wherein the cavity is a defect in a photonic band-gap structure.
 43. The method of claim 31, wherein the cavity is a superconductor structure.
 44. The method of claim 31, wherein the electromagnetic field includes a component generated by one or more laser beams.
 45. The method of claim 31, wherein the electromagnetic field includes a component generated by an externally applied magnetic field.
 46. The method of claim 31, wherein the semiconductor quantum bits are vertically coupled quantum dots.
 47. The method of claim 31, wherein the semiconductor quantum bits are horizontally coupled quantum dots.
 48. A quantum computing apparatus, comprising: a cavity excited by an electromagnetic field, wherein multiple semiconductor quantum bits comprising electron-doped quantum dots are located in the cavity; and means for effecting a controlled NOT (CNOT) operation involving any pair of quantum dots by tuning energy levels of the quantum dots into resonance with frequencies of the cavity.
 49. The apparatus of claim 48, wherein the energy levels of the quantum dots are tuned by voltages applied to gates across the quantum dots.
 50. The apparatus of claim 48, wherein the energy levels of the quantum dots are tuned by pulses of electromagnetic radiation focused onto the quantum dots.
 51. The apparatus of claim 48, wherein the semiconductor quantum bits are arranged in an array.
 52. The apparatus of claim 48, wherein each of the semiconductor quantum bits is a quantum dot doped with a single electron.
 53. The apparatus of claim 52, wherein the information is represented by the states of an electron trapped in the quantum dot.
 54. The apparatus of claim 53, wherein the state of the electron is determined by detecting selective fluorescent emissions from the trapped electron.
 55. The apparatus of claim 53, wherein the information is contained in a spin-state of the electron.
 56. The apparatus of claim 55, further comprising means for reading the information by determining the spin-state of the trapped electron.
 57. The apparatus of claim 56, wherein the spin-state is determined by detecting selective fluorescent emissions from the trapped electron.
 58. The apparatus of claim 48, wherein the cavity is a whispering-gallery-mode resonator.
 59. The apparatus of claim 48, wherein the cavity is a defect in a photonic band-gap structure.
 60. The apparatus of claim 48, wherein the cavity is a superconductor structure.
 61. The apparatus of claim 48, wherein the electromagnetic field includes a component generated by one or more laser beams.
 62. The apparatus of claim 48, wherein the electromagnetic field includes a component generated by an externally applied magnetic field.
 63. The apparatus of claim 48, wherein the semiconductor quantum bits are vertically coupled quantum dots.
 64. The apparatus of claim 48, wherein the semiconductor quantum bits are horizontally coupled quantum dots. 